Packing the Hypercube

Abstract Let G be a graph that is a subgraph of some n-dimensional hypercube Qn. For sufficiently large n, Stout [20] proved that it is possible to pack vertex- disjoint copies of G in Qn so that any proportion r < 1 of the vertices of Qn are covered by the packing. We prove an analogous theorem for edge-disjoint packings: For sufficiently large n, it is possible to pack edge-disjoint copies of G in Qn so that any proportion r < 1 of the edges of Qn are covered by the packing.

[1]  Ran Libeskind-Hadas,et al.  On Edge-Disjoint Spanning Trees in Hypercubes , 1999, Inf. Process. Lett..

[2]  Yuan Zhou.  Introduction to Coding Theory , 2010 .

[3]  Mark Ramras.  Symmetric edge-decompositions of hypercubes , 1991, Graphs Comb..

[4]  John Frederick Fink.  On the decomposition of n-cubes into isomorphic trees , 1990, J. Graph Theory.

[5]  Frank Thomson Leighton.  Introduction to parallel algorithms and architectures: arrays , 1992 .

[6]  David W. Krumme,et al.  Fixed Hypercube Embedding , 1987, Inf. Process. Lett..

[7]  Dean G. Hoffman,et al.  Star Decompositions of Cubes , 2001, Graphs Comb..

[8]  Ron Holzman,et al.  Packing lines in a hypercube , 1993, Discret. Math..

[9]  Stephan G. Wagner,et al.  Decomposing the hypercube Qn into n isomorphic edge-disjoint trees , 2012, Discret. Math..

[10]  M. Livingston,et al.  Embeddings in hypercubes , 1988 .

[11]  F. Harary,et al.  Covering and packing in graphs—V: Mispacking subcubes in hypercubes , 1988 .

[12]  Frank Thomson Leighton,et al.  Hamming Codes, Hypercube Embeddings, and Fault Tolerance , 2007, SIAM J. Comput..

[13]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[14]  Mark Ramras.  Symmetric vertex partitions of hypercubes by isometric trees , 1992, J. Comb. Theory, Ser. B.

[15]  D. Djoković.  Distance-preserving subgraphs of hypercubes , 1973 .

[16]  F. Harary,et al.  A survey of the theory of hypercube graphs , 1988 .

[17]  Michael Randolph Garey,et al.  On cubical graphs , 1975 .

[18]  Mark Ramras.  Fundamental Subsets of Edges of Hypercubes , 1997, Ars Comb..

[19]  Edgar M. Palmer,et al.  On the spanning tree packing number of a graph: a survey , 2001, Discret. Math..